Parthasarathy's Dirac operator inequality [1]} is a valuable tool on this aspect. Indeed, let \(\theta \)  be a Cartan involution of \(G\)  and assume that \(K:=G^{\theta }\)  is a maximal compact subgroup of \(G\) . Let
\(\mathfrak {g}_0=\mathfrak {k}_0 \oplus \mathfrak {p}_0\) 
